Finite Group Databasebeta

← #27151128

Subgroups — #27151128

Order 12416.

766 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 161 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261242C213yesmaximal, complemented
2#20260925C214yesmaximal, complemented
3#20260922C214yesmaximal, complemented
4#20260709C213yesmaximal, complemented
5#585×C97C211yesmaximal, characteristic, complemented
6C22×C4×D97C2211yescharacteristic, complemented
7#15690791C2216yescomplemented
8#15690786—23no—
9#15690781—23no—
10#15690778C2216yescomplemented
11C4×C22⋊D97—26no—
12#15690769C2213yescomplemented
13#15690725C22112yescomplemented
142−1+4×C97—23no—
15H(5,𝔽2)×C97—23no—
16H(5,𝔽2)×C97C2214yes—
17C2×#41×C97—26no—
18C2×#41×C97C2213yes—
19C23×D97C2311yescharacteristic
20C2×C22⋊D97—26no—
21C22×Dic97C2311yescharacteristic
22C2×C4×D97—23no—
23C2×C4×D97—23no—
24C4⋊Dic97—23no—
25#13095490—26no—
26C22⋊Dic97—26no—
27C22⋊Dic97C2316yes—
28#13095488—26no—
29C4×Dic97—23no—
30#41×C97—24no—
31#41×C97—212no—
32#41×C97—26no—
33#41×C97—212no—
34#41×C97—26no—
35C2×Q8×C97—26no—
36C2×Q8×C97D413yescomplemented
37C2×D4×C97—26no—
38C2×D4×C97—212no—
39C2×D4×C97D413yescomplemented
40C2×D4×C97C2316yes—
41C22×C4×C97—24no—
42C22×C4×C97D416yescomplemented
43C22×C4×C97C2311yescharacteristic
44C23×C97—24no—
45C23×C97C2×D416yes—
46C23×C97C2411yescharacteristic
47C22×D97—23no—
48C22×D97—42no—
49Q8×C97—24no—
50Q8×C97—26no—
51D4×C97—212no—
52D4×C97—212no—
53D4×C97—26no—
54C2×C4×C97—212no—
55C2×C4×C97—26no—
56C2×C4×C97—26no—
57C2×C4×C97C2×D416yes—
58C2×C4×C97C2×D413yes—
59C2×C4×C97C2×D413yes—
60C22⋊D97—412no—
61C2×Dic97—26no—
62C2×Dic97—23no—
63C4×D97—43no—
64C4×D97—41no—
65C22×C97—26no—
66C22×C97—41no—
67C22×C97—46no—
68C22×C97—84no—
69C22×C97C22×D413yes—
70C2×D97—26no—
71C2×D97—43no—
72C2×D97—41no—
73C4×C97—26no—
74C4×C97#12013yes—
75C4×C97#12011yescharacteristic
76Dic97—46no—
77Dic97—41no—
78C2×C97—23no—
79C2×C97—46no—
80C2×C97C2×#12011yescharacteristic
81D97—42no—
82#3025—971nomaximal, Sylow
83C97#302511yescharacteristic, complemented, Sylow
84#585D9711yescharacteristic, complemented
85#525—973no—
86#458—974no—
87B(4,𝔽2)—974no—
88#410—973no—
892−1+4—23no—
90H(5,𝔽2)—23no—
91H(5,𝔽2)C2×D9714yes—
92C2×#41—26no—
93C2×#41C2×D9713yes—
94C23×C4—971no—
95#123—976no—
96C22⋊Q8—1943no—
97#121—1943no—
98#120—976no—
99C4×D4—1946no—
100#117—973no—
101#99—9712no—
102C24—971no—
103#41—24no—
104#41—212no—
105#41—26no—
106#41—212no—
107#41—26no—
108C2×Q8—26no—
109C2×Q8C22⋊D9713yescomplemented
110C2×D4—26no—
111C2×D4—212no—
112C2×D4—1946no—
113C2×D4C22⋊D9713yescomplemented
114C2×D4C22×D9716yes—
115C22×C4—24no—
116C22×C4—971no—
117C22×C4—1943no—
118C22×C4—1943no—
119C22×C4C22⋊D9716yescomplemented
120C22×C4C22×D9711yescharacteristic
121C4⋊C4—1946no—
122C4⋊C4—1943no—
123C22⋊C4—976no—
124C22⋊C4—1946no—
125C22⋊C4—1946no—
126C42—1943no—
127C23—24no—
128C23—3882no—
129C23—1943no—
130C23C2×C22⋊D9716yes—
131C23C23×D9711yescharacteristic
132Q8—24no—
133Q8—26no—
134D4—212no—
135D4—212no—
136D4—26no—
137D4—38812no—
138C2×C4—212no—
139C2×C4—26no—
140C2×C4—26no—
141C2×C4—3883no—
142C2×C4—3881no—
143C2×C4—1946no—
144C2×C4—1943no—
145C2×C4C2×C22⋊D9716yes—
146C2×C4C2×C22⋊D9713yes—
147C2×C4C2×C22⋊D9713yes—
148C22—26no—
149C22—41no—
150C22—46no—
151C22—84no—
152C22—3883no—
153C22—3881no—
154C22—1946no—
155C22C22×C22⋊D9713yes—
156C4—26no—
157C4—3886no—
158C4—3881no—
159C4#1569077913yes—
160C4#1569077911yescharacteristic
161C2—23no—
162C2—46no—
163C2—3882no—
164C2C2×#1569077911yescharacteristic
subgroupG/H
centerC4#15690779
commutator, cosocleC23×C97C24
FrattiniC23C23×D97
Fitting#585×C97C2
socleC2×C97C2×#120
hypercenter#585D97
hypocenterC97#3025
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C22×C4 ◁ #585

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #20261242C2semidirect
2. #20260925C2semidirect
3. #20260922C2semidirect
4. #20260709C2semidirect
5. #585×C97C2semidirect
6. C22×C4×D97C22semidirect
7. #15690791C22semidirect
10. #15690778C22semidirect
12. #15690769C22semidirect
13. #15690725C22semidirect
36. C2×Q8×C97D4semidirect
39. C2×D4×C97D4semidirect
42. C22×C4×C97D4semidirect
83. C97#3025semidirect
84. #585D97semidirect
109. C2×Q8C22⋊D97semidirect
113. C2×D4C22⋊D97semidirect
119. C22×C4C22⋊D97semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202612422
#202609252
#202609222
#202607092
#585×C972
#302597